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Annuity Markets and Capital Accumulation

Shantanu Bagchi and James A. Feigenbaum

Abstract

We examine how the absence of annuities in financial markets affects capital accumulation in a two-period overlapping generations model. Our findings indicate that the effect on capital is ambiguous in general equilibrium, because there are two competing mechanisms at work. On the one hand, the absence of annuities increases the price of old-age consumption relative to the price of early-life consumption. This induces a substitution effect that reduces saving and capital, and an income effect that has the opposite effect as households want to consume less when young, causing them to save more. On the other hand, accidental bequests originate from the assets of the deceased under missing annuity markets. The bequest received in early life always has a positive income effect on saving, but the bequest received in old age, conditional on survival, is effectively a partial annuity with both substitution and income effects. We find that when the desire to smooth consumption is high, the income effects dominate, so the capital stock always increases when annuity markets are missing. However, when the desire to smooth consumption is low, the substitution effects dominate, and the capital stock decreases with missing annuity markets.

JEL Classification: D15; D52; E21

Appendix

A The Social Planner’s Problem

Suppose that we have a social planner who maximizes

(26)V=t=0ρt[u(ct,0)+βQu(ct+1,1)]

for some sequence of Pareto weights ρt ≥ 0, subject to the feasibility constraint that aggregate consumption and next period’s capital stock equal current output and any undepreciated capital

(27)ct,0+Qct,1+kt+1=f(kt)+(1δ)kt

and given both k0 > 0 and c0,1 > 0.

The Lagrangian for the social planner’s problem is

(28)LV=t=0{ρt[u(ct,0)+βQu(ct+1,1)]+λt[f(kt)+(1δ)ktct,0Qct,1kt+1]},

which generates the first-order conditions

(29)LVct,0=ρtu(ct,0)λt=0t0,
(30)LVct+1,1=ρtβQu(ct+1,1)λt+1Q=0t0,

and

(31)LVkt+1=λt+1[f(kt+1)+1δ]λt=0t0.

Combining these first-order conditions, we obtain the usual Euler equation

(32)u(ct,0)=β[f(kt+1)+1δ]u(ct+1,1)

that applies in the absence of mortality risk. A steady state allocation, for which kt=k, ct,0=c0, and ct,1=c1 for all t ≥ 0, will be optimal with

(33)V=u(c0)+βQu(c1)11f(k)+1δ

if

(34)ρtρt+1=λtλt+1=f(k)+1δ>1

for all t ≥ 0; i. e. if the allocation is dynamically efficient.[9] Thus for any k* that satisfies (34), we will have a stationary Pareto optimal allocation that satisfies

(35)u(c0)=β[f(k)+1δ]u(c1)

and

(36)c0+Qc1=f(k)δk.

Comparing (35) to (8) and (36) to (10), we see that the steady state competitive equilibrium in the annuities regime will be a stationary Pareto optimal allocation if ka satisfies (34). That is to say, given k0=ka and c1,0=c1, there can be no feasible allocation {ct,0,ct+1,1}t=0 for which

u(ct,0)+βQu(ct+1,1)u(c0)+βQu(c1)

for all t with strict inequality for some t. Otherwise, such an allocation would yield V > V* when V* maximizes (26) over the set of feasible allocations.[10]

In contrast, if we compare (35) to (16), assuming Q < 1 these two equations will not be the same. Thus a competitive equilibrium in the bequest regime will not be Pareto optimal. Starting from the bequest-regime competitive equilibrium, there will be Pareto-improving transitions that go to steady states with higher steady state utility than the lifetime utility in the bequest regime. This does not, however, imply that it is possible to transition from the bequest-regime competitive equilibrium to the annuity-regime competitive equilibrium in a Pareto-improving fashion. There are infinitely many Pareto optimal steady state allocations. Only one of these corresponds to the annuity-regime competitive equilibrium. All of the other steady state allocations can only be achievable in a market setting if there are exogenous transfers. Households may do better in a competitive equilibrium with bequests than in a competitive equilibrium with perfect annuities if the bequest is a second-best mechanism that approximates a transfer which can yield a bigger lifetime utility if the presence of perfect annuity markets. For more on this result, see Bagchi and Feigenbaum (2019).

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Published Online: 2019-11-28

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